Simplify; express your answer in exponential form. Assume $y\neq 0, a\neq 0$. $\dfrac{{(y^{-3}a^{-2})^{-5}}}{{(y^{2}a^{-3})^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-3}a^{-2})^{-5} = (y^{-3})^{-5}(a^{-2})^{-5}}$ On the left, we have ${y^{-3}}$ to the exponent ${-5}$ . Now ${-3 \times -5 = 15}$ , so ${(y^{-3})^{-5} = y^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-3}a^{-2})^{-5}}}{{(y^{2}a^{-3})^{-5}}} = \dfrac{{y^{15}a^{10}}}{{y^{-10}a^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{15}a^{10}}}{{y^{-10}a^{15}}} = \dfrac{{y^{15}}}{{y^{-10}}} \cdot \dfrac{{a^{10}}}{{a^{15}}} = y^{{15} - {(-10)}} \cdot a^{{10} - {15}} = y^{25}a^{-5}$